Mutual Multilinearity of Nonequilibrium Network Currents

TL;DR

This work proves that stationary Currents $\{\jmath_e\}$ in nonequilibrium Markov networks exhibit mutual multilinearity (MML) when multiple input edges have tunable transition rates. It provides two independent proofs—a graph-theoretic, Kirchhoff-current-law–based induction and a Laplace-space resolvent approach—yielding explicit current-to-current susceptibilities $\lambda^{\mathcal{I}}_{e\leftarrow i}$ that map dependencies across the network. When the input set is a full fundamental set, MML reduces to Kirchhoff’s current law, and with equilibrium references it connects to equilibrium fluctuations via a linear response formula $\jmath_e = (c^{\mathrm{eq}}_{e i}/c^{\mathrm{eq}}_{i i})\jmath_i$, with a multivariable extension for several inputs. The results rely on spanning-tree polynomials and a careful treatment of admissible input sets, and they generalize previous single-input results to arbitrary network topologies and input sizes, enabling explicit control and prediction of current reconfiguration in nonequilibrium networks.

Abstract

Continuous-time Markov chains have been successful in modelling systems across numerous fields, with currents being fundamental entities that describe the flows of energy, particles, individuals, chemical species, information, or other quantities. They apply to systems described by agents transitioning between vertices along the edges of a network (at some rate in each direction). It has recently been shown by the authors that, at stationarity, a hidden linearity exists between currents that flow along edges: if one controls the current of a specific "input" edge (by tuning transition rates along it), any other current is a linear-affine function of the input current [PRL 133, 047401 (2024)]. In this paper, we extend this result to the situation where one controls the currents of several edges, and prove that other currents are in linear-affine relation with the input ones. Two proofs with distinct insights are provided: the first relies on Kirchhoff's current law and reduces the input set inductively through graph analysis, while the second utilizes the resolvent approach via a Laplace transform in time. We obtain explicit expressions for the current-to-current susceptibilities, which allow one to map current dependencies through the network. We also verify from our expression that Kirchhoff's current law is recovered as a limiting case of our mutual linearity. Last, we uncover that susceptibilities can be obtained from fluctuations when the reference system is originally at equilibrium.

Figures (4)

• Figure 1: Illustration of central graph-theoretical notions. Starting from a connected graph (left panel), a set of edges is identified in red (middle panel) whose removal results in a spanning tree (right panel), a connected graph without cycles that contains all vertices. Adding back one chord to the spanning tree creates exactly one cycle.
• Figure 2: Summarizing two types of mutual linearity among nonequilibrium stationary currents. (Left) Any stationary current $\jmath_e$ (blue) in the graph can be represented as a linear-affine function of an "input" current $\jmath_i$ (red) when the input edge $i$ is controlled; the scalar coefficients $\jmath^{\smallsetminus i}_{e}$ and $\lambda^{i}_{e \leftarrow i}$ depend both on dynamical and topological properties of the Markov process. This result expresses the mutual linearity of currents, derived in Ref. harunari2024mutual. (Right) Kirchhoff's current law: A fundamental set of edges (red) whose removal leaves a connected graph without cycles is fixed. The stationary current of any other edge (blue) can always be represented as a linear combination of the currents along the red edges (including when these edges are controlled). These choices of edges are not unique, and the coefficients $\gamma_{e,i}$ are either 0 or $\pm 1$ depend solely on the graph topology. In this paper, we address the intermediate situation in which more than one edge is controlled, but KCL does not hold yet.
• Figure 3: Illustration of the admissibility concept. Top: The removal of the input edges results in a connected graph, satisfying the criterion for admissibility. Bottom: The resulting graph is disconnected thus the input set is not admissible.
• Figure 4: Resulting graph after removal of $\mathcal{I}$ and $e$ Left: $e$ and $i$ do not belong to the same cycle $\mathscr{C}$, notice that $i$ is fully contained by $\mathscr{T}^-$. Right: Since $e$ and $i$ belong to the same cycle, $i$ is in between both trees $\mathscr{T}^\pm$.